Introduction to Probability with Texas Hold 'em Examples

Frederic Paik Schoenberg

• Éditeur: Chapman and Hall/CRC
• Année: 2016
• ISBN:
  9781351737746
• Édition: 2
• Sujet: Mathematics
• Sous-sujet: Games in Mathematics

Introduction to Probability with Texas Hold'em Examples illustrates both standard and advanced probability topics using the popular poker game of Texas Hold'em, rather than the typical balls in urns. The author uses students' natural interest in poker to teach important concepts in probability.

CHAPTER 1

Probability Basics

1.1 Meaning of Probability

It’s hand 229 of day 7 of the 2006 World Series of Poker (WSOP) Main Event, the biggest poker tournament ever played, and after the elimination of 8770 entrants it is now down to the final three players. The winner gets a cash prize of $12 million, second place just over $6.1 million, and third place gets about $4.1 million. Jamie Gold is the chip leader with $60 million in chips, Paul Wasicka has $18 million, and Michael Binger has $11 million. The blinds are $200,000 and $400,000, with $50,000 in antes. (Note: For readers unfamiliar with Texas Hold’em, please consult Appendices A and B for a brief explanation of the game and related terminology used in this book.) Gold calls with 4♠ 3♣, Wasicka calls with 8♠ 7♠, Binger raises with A♥ 10♥, and Gold and Wasicka call. The flop is 10♣ 6♠ 5♠. Wasicka checks, perhaps expecting to raise, but by the time it gets back to him, Binger has bet $3.5 million and Gold has raised all-in. What would you do, if you were Paul Wasicka in this situation?

There are, of course, so many issues to consider. One relatively simple probability question that may arise is this: given all the players’ cards and the flop, if Wasicka calls, what is the probability that he will make a flush or a straight?

This is the type of calculation we will address in this book, but before we get to the calculation, it is worth examining the question a bit. What does it mean to say that the probability of some event, like Wasicka’s making a flush or a straight, is, say, 55%? It may surprise some readers to learn that considerable disagreement continues among probabilists and statisticians about the definition of the term probability. There are two main schools of thought.

The frequentist definition is that if one were to record observations under the exact same conditions over and over, with each observation independent of the rest, then the event in question would ultimately occur 55% of the time. In other words, to say that the probability of Wasicka making a flush or a straight is 55% means that if we were to imagine repeatedly observing situations just like this one, or if we were to imagine dealing the turn and river cards repeatedly, each time reshuffling the remaining 43 cards before dealing, then Wasicka would make a flush or a straight 55% of the time.

The Bayesian definition is that the quantity 55% reflects one’s subjective feeling about how likely the event is to occur: in this case, because the number is 55%, the feeling is that the event is slightly more likely to occur than it is not to occur.

The two definitions suggest very different scientific questions, statistical procedures, and interpretations of results. For instance, a Bayesian may discuss the probability that life exists on Mars, while a frequentist may argue that such a construct does not even make sense. Frequentists and Bayesians have had spirited debates for decades on the meaning of probability, and consensus appears to be nowhere in sight.

While experts may differ about the meaning of probability, they have achieved unanimous agreement about the mathematics of probability. Both frequentists and Bayesians agree on the basic rules of probability—known as the axioms of probability. These three axioms are discussed in Section 1.3, and the methods for calculating probabilities follow from these three simple rules. There is also agreement about the proper notation for probabilities: we write P(A) = 55%, for instance, to connote that the probability of event A is 55%.

1.2 Basic Terminology

Before we get to the axioms of probability, a few terms must be clarified. One is the word or. In the italicized question about probability in Section 1.1, it is unclear whether we mean the probability that Wasicka makes either a flush or a straight but not both or the probability that Wasicka makes a flush or a straight or both. English is ambiguous regarding the use of or. Mathematicians prefer clarity over ambiguity and have agreed on the convention that A or B always means A or B or both. If one means A or B but not both, one must explicitly include the phrase but not both.

Of course, in some situations, A and B cannot both occur. For instance, if we were to ask what the probability is that Wasicka makes a flush or three 8s on this hand, it is obvious that they cannot both happen: if the turn and river are both 8s, then neither the turn nor the river can be a spade, since Wasicka already has 8♠. If two events A and B cannot both occur, i.e., if P(A and B) = 0, then we say that the events are mutually exclusive. We usually use the notation AB to denote the event A and B, so the condition for mutual exclusivity can be written simply P(AB) = 0.

The collection of all possible outcomes is sometimes called the sample space, and an event is a subset of elements of the sample space. For instance, in the WSOP example described in the beginning of Section 1.1, if one were to consider the possibilities that might come on the turn, one might consider the sample space to be all 52 cards. Of course, if we know what cards have been dealt to the three players and which cards appeared on the flop, then these nine cards have zero probability of appearing on the turn, and we may consider the remaining 43 cards equally likely. The event that the turn is the 7♦ is an event consisting of a single element of the sample space, and the event that the turn is a diamond is an event consisting of 13 elements.

Given an event A, we use the notation A^c to mean the complement of A, or in other words the event that A does not occur. If A is the event that the turn is a diamond, for instance, then A^c is the event that the turn is a club, heart, or spade. For any event A, the events A and A^c are always mutually exclusive and exhaustive, meaning that together they cover the entire sample space.

1.3 Axioms of Probability

Three basic rules or axioms must govern all probabilities:

1. P(A) ≥ 0.
2. P(A) + P(A^c) = 1.
3. If A₁, A₂, A₃, … are mutually exclusive events, then P(A₁ or A₂ or … A_n)...